My name is Stephen Ren, I am currently a high school student, and I have always loved mathematics. Several months ago, I learned of the connection between complex numbers and fractals, and wrote a Java program to try it out for myself. I had recently been tackling an algebra problem regarding tetration, and decided to play around with tetration fractals using my program. After producing the standard tetration fractal, I decided to experiment. I changed the representation of the fractal from convergence/divergence of the point to how "steep" the point gets as tetration was applied (explained in greater detail later), and to my surprise, produced very interesting and beautiful fractals, several of which I have attached to this post. Many have been resized to fit the restraints of the post. For full sized images I just created an Imgur album yesterday at https://imgur.com/a/D5g78, and I will be adding more and more images to it as time goes on.

I have not been able to find these fractals or style of fractals anywhere else, which is a shame. The closest I have come to finding one is called Pickover Stalks, which is based on the distance the point gets to the imaginary axis. Though similarities are apparent, it is not quite the same, and I had not found examples of it with tetration fractals. I feel this is kind of like a "hidden gem" for tetration fractals, as I applied the same method to the Mandelbrot set and the result was very, very underwhelming. In any case, I seems to me that tetration fractals are often overlooked when compared to the Mandelbrot, Julia sets, and the like, and I wish this may at least do something to change that.

As for the generation of these fractals:

Normal tetration fractals are generated by coloring each point based on convergence/divergence. However, for these fractals, each pixel is colored based on the maximum reference angle as tetration was applied to the point it represents in the complex plane. For example (and I apologize for any errors in notation I may have made), if F^n(z) = z^(z^(z^(z ... n times))) represents tetration, the color of the pixel representing a point z would be based on the maximum value in the set of the reference angles for arg(F^k(z)) for k = 1 ... n, where n was the number of iterations of tetration (as with all fractals, more is needed to see detail at greater zoom, for these images n ranged from 100 to 400). At first I started k at zero but as you would expect all it does is overlay a gradient over the fractal, otherwise it looked exactly the same.

Conceptually, it is simply colored based on how "steep" the point gets along its orbit.

For reference, the following image is of this fractal centered on the origin, with the real axis ranging from -6 to 6. This was generated with 100 iterations of tetration

For comparison, this next image is of "normal" convergence/divergence tetration fractals with the same center and scale.

The following images are several more portions of the fractal, all of which are zoomed in snapshots of the first image. I hope you all enjoy.

This image is centered on -4.125, magnified 40x from the original fractal. The overall shape of this figure is just barely visible in the complete depiction, but, as shown in this image, it is heavily laced with detail. This figure is particularly interesting because it is set apart from the rest of the “main” fractal body, and further so far I had not found it repeated elsewhere in the fractal. This image was created with 100 iterations of tetration.

This image is centered on 0.9073 + 2.354i, and is magnified 50x from the original fractal. This figure shows up repeatedly within the fractal, not only in isolation, but often within in other patterns and figures. I think there was a post on here about Kneser’s Chi-Star function? I don't quite understand what was but the pattern bears some resemblance to this, though it may be coincidence. This image was created with 100 iterations of tetration.

This image is centered on 0.170938545, and is magnified 6000x from the original fractal. Patterns similar to this can be found repeated in many areas along the real axis, though it is often deeply burred in a larger figure. This image is generated with 400 iterations of tetration.

Please do note, however, that I have only a basic understanding in tetration, though I have a decent amount of math knowledge for a high schooler (I was 1 point away from qualifying for usamo). I am afraid I may have to leave much of the theoretical heavy lifting to all of you, however if possible, I am always looking to be able to learn more things. If you have any questions or insights feel free to email me or comment on this post.

If any of you are curious, this is the story of how I came across these fractals, if any of you are interested (it is quite long):

I had a math problem that was an age old nemesis of mine. When I first learned about power towers (which I now know as tetration) in the sixth grade, I was shocked at how ridiculously quickly the numbers grew. I figured there was no way that 2^(2^(2^(2^2))) was as large a number as my teacher claimed it to be, nor that adding another two to that expression would make its decimal representation too large to write down given the physical constraints of the universe. After a little computation, I found myself humbled and intrigued.

After that, I would occasionally play with power towers in my calculator, mainly because I liked to see it trip over itself with overflow errors. Eventually, I picked a small enough number as a base, and was quickly disappointed as no matter how many times I applied the tetration it would never seem to get anywhere. After a lot of experimentation, I found that 1.44 converged while 1.45 grew off to infinity, and the same for 1.444 and 1.445, respectively. However, not very surprisingly, this did not hold for 1.4444 and 1.4445. It was at this point that I wondered what the largest number such that repeated tetration would converge. I had an elementary understanding of limits at the time, but it did not extend much beyond things such as "the limit of x^2 as x -> 3 was 9", so I figured they were quite useless and common sense worked as well or better most of the time. I also did not know calculus, so I never solved that problem until I revisited it this year.

For the longest time I figured I would never find the answer to this problem (I did not learn calculus until 10th grade). Once in a while, I would sometimes think about this problem as I pondered the other mysteries of life, and even after I learned calculus I didn't think to apply it. However, one day, I finally, finally had a breakthrough. I tried visualizing the problem graphically, with the equation y = n ^ x representing the tetration, and the line y = x to change the output of the function into the new input. For example if you had n = 2, first you would start at x = 1 on the exponential function, giving you the point (1, 2). Then, you "move" horizontally until you hit y = x, so that now your x value was your previous y value, in this case two. Next you move vertically up so that you hit 2 ^ x again, and since this time x was the output of the previous iteration, you have reached the next step in the power tower. This "staircase"-like process repeated over and over represents the creation of higher and higher tetrations.

The solution was immediately obvious, what I was looking for was an n such that n ^ x was tangent to y = x. Solving the system, the answer comes out to be n = e ^ 1/e ~= 1.4447. I was quite ecstatic. I didn't know the term "tetration" at the time, but of course, the Wikipedia article on tetration mentions that Euler had proved this conjecture. Though honestly I am glad I didn't find that Wikipedia page until later, because the sense of accomplishment and discovery at the point of finding the answer was amazing.

How did this tie into fractals? Well during that week, aside from solving this problem that haunted me for several years, I developed an interest in complex fractals (I clicked on a Mandelbrot zoom on Youtube for some reason, and became intrigued). Astonishingly, two days after I solved my problem, I stumbled into this Mathologer video: https://www.youtube.com/watch?v=9gk_8mQuerg , and the visualization at 8:55 was almost exactly how I had solved my problem. I was very, very much intrigued. That ultimately led me to try to create my own fractals based on tetration (I had been considering writing a fractal generating program, but at that moment I dropped everything and began working).

As I was working on my program, the first step was defining an imaginary class for objects that acted like imaginary numbers. Everything was straightforward enough, however, I soon ran into a problem - complex exponents, which formed the basis of tetration fractals.

I figured the way to do this was through Demoivre’s, by turning the base complex number into exponential form and keeping the exponent in rectangular form and simplifying from there. It turned out to be simple algebra, but, to me at least, it was an "ooooh" moment, as though I had previously used Demoivre's for solving a variety math problems, I never really dealt with complex powers, but seeing it applied this way made me appreciate even more how flexible it was. After making sure it works by verifying it with several online sources and testing with many different complex numbers, I continued with the rest of the program, which was more or less straightforward.

Once I finally got rid of all the random bugs and had my program up and running, it was pretty late into the night, but I was extremely excited when the tetration fractal showed up on my screen (these were the normal tetration fractals, not the new kind I stumbled across). For the next few days I played around with the fractal in my spare time, and implemented something akin to a zoom method.

Then I started to experiment. My main goal when I went into this was wanting to create tetration fractals, and though I was extremely happy that it worked out, after the initial euphoria I started to wish they looked a just little bit... better. So I messed with the things that gave the Mandelbrot and Julia sets their iconic look, such as escape and period. Eventually, I tried changing the representation from convergence/divergence to the angle of the point as tetration is applied, not knowing what to expect. This is where things got interesting.

These images are very stunning, and you could imagine my shock when they first shown through, especially contrasted with the rather bland style of normal tetration fractals in black and white. Of course, when I first used this method, the mapping of angle to color was linear, and the fractal was in a sense a lot thicker and darker, but the overall shape was there and it was different enough that I dove into it. I played around with converting between angle and color with a straightforward function. The current mapping I use for these images is based on 1/x.

Here, the x represents one minus the ratio of theta (the maximum angle previously mentioned) to pi/2. On the y scale, 1 represents an rgb of (255, 255, 255), and 0 is (0,0,0), with a linear relationship in between.

Though you use simple algebraic transformation to modify the "thickness" of the branches in the fractal, the construction of the fractal itself is based just on tetration and argument of the point, and no complex algorithms such as fractal overlaying or anything like that was used, which is why I'm surprised it came out as detailed as it did.

(09/02/2017, 12:10 AM)stephrenny Wrote: If any of you are curious, this is the story of how I came across these fractals ...

Nice pictures; thanks for posting. I am familiar with the black and white escape charts (from Daveney's work) showing the fractal for iterated exponentiation, which corresponds to your 2nd image. Your 2nd image shows the Shell Thron region, which is the largest interior white area . I don't understand where the interior lines inside the Shell Thron region in your 1st image come from. Are they artifacts? Welcome to the forum.